Direct-product-closed subgroup property

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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Definition

Definition with symbols

A subgroup property $p$ is said to be direct-product-closed if whenever $H_{1}\leq G_{1}$ satisfies $p$, and $H_{2}\leq G_{2}$ satisfies $p$, then $H_{1}\times H_{2}$ satisfies $p$ as a subgroup of $G_{1}\times G_{2}$.

Relation with other metaproperties

Stronger metaproperties

• Intersection-closed subgroup property that also satisfies inverse image condition

Weaker metaproperties

• Direct-power-closed subgroup property

Related metaproperties

• Subgroup properties satisfying image condition
• Subgroup properties satisfying inverse image condition
• Subgroup properties satisfying transfer condition
• Subgroup properties satisfying intermediate subgroup condition